THE MODELLING AND CONTROL OF AN AUTOMOTIVE DRIVETRAIN

Transcription

1 THE MODELLNG AND CONTROL OF AN AUTOMOTVE DRVETRAN Thei preented in partial ulilment o the requirement or the degree MASTER OF SCENCE N ENGNEERNG By Nihola M. Northote Supervior Dr. A.B. Taylor Department o Mehanial Engineering Univerity o Stellenboh Co Supervior Mr. J. Treurniht Department o Eletrial and Eletroni Engineering Univerity o Stellenboh April 6

2 Delaration Delaration, the underigned, delare that the work ontained in thi thei i my own original work and ha not previouly, in it entirety or in part, been ubmitted at any univerity or a degree. Signature o Candidate Date i

3 Abtrat Abtrat Shunt and hule in a vehile drivetrain are two driveability related phenomena reponible or driver diomort. They are experiened a a harp jerk (hunt) ollowed by a erie o longitudinal oillation (hule) and are indued by a rapid hange in engine torque. The ue o drive by wire throttle in modern day vehile enable the onboard eletroni ontrol unit to manipulate the driver torque demand beore ending a revied torque demand ignal to the engine. n thi way a eedbak ontrol ytem an be ued to enure that the drivetrain ollow the driver torque demand a quikly a poible without induing hunt or hule. n thi projet a drivetrain model wa derived and it parameter experimentally determined. The auray o the model wa validated uing tet data rom a vehile, and the onluion wa made that the model wa an aurate vehile imulation tool. A drivetrain ontroller wa then deigned and it perormane imulated uing the vehile model. The imulation howed that the ontroller igniiantly redued the hunt and hule in the drivetrain thereby improving driver omort. ii

4 Samevatting Samevatting Shunt en hule in n voertuig i n ooraak van betuurderongemak. Hierdie verkynel word ondervind a n kerp rukbeweging ( hunt ) gevolg deur n reek horiontale oorgangverkynel ( hule ), en gebeur a gevolg van n kielike verandering in enjindraaimoment. Die gebruik van drive by wire moorkleppe in moderne voertuie maak dit moontlik vir die motor e elektroniee beheereenheid om die betuurder e draaimomentvereite te wyig. n Heriende draaimoment aanvraag kan dan na die enjin getuur word. n Terugvoerbeheertelel kan du gebruik word om te vereker dat die drylyn die draaimomentvereite van die betuurder o vinnig a moontlik al volg, onder om hunt o hule te verooraak. n hierdie projek i n drylyn model agelei en y ekperimentele parameter bepaal. Data akomtig van toete op n voertuig i gebruik om die akuraatheid van die model te bevetig. Die gevolgtrekking i gemaak dat die model n akkurate voertuig imulaiemiddel i. n Drylyn beheerder i ontwerp en die pretaie i geimuleer. Die imulaie het gewy dat die beheerder die hunt en hule in die drylyn aanienlik verminder het en gevolglik i betuurdergemak verbeter. iii

5 Aknowledgement Aknowledgement Firt o all would like to thank my upervior Dr. Andrew Taylor or teahing me to earh or reaon why dream an be ahieved rather than or reaon why they an t. Thank you alo to my o upervior Mr. Johan Treurniht or all hi patiene and guidane over the lat year. A peial mention o thank goe to Cobu and Ferdi Zietman or all their help with the experimental work in thi thei. Finally on a more peronal note, am deeply grateul to my parent George and Brunella. Thank you or your never ending love and upport, and or teahing me the bai priniple o honety and hard work. iv

6 Content Content Delaration Abtrat Samevatting Aknowledgement Content Lit o Figure Lit o Table Gloary Chapter : ntrodution i ii iii iv v viii x xi.. Projet Bakground Projet Objetive 3.3 Projet Outline 3.4 Chapter Overview 4 Chapter : Literature Review 5. ntrodution 5. Vehile Driveability 5.3 Drivetrain Modelling 6.4 Drivetrain Control 8 Chapter 3: Drivetrain Modelling 3. ntrodution 3. Complex Model 3.. Engine, Flywheel and Cluth 3.. Tranmiion and Drivehat Wheel and Vehile Summary Simpliied Model 6 Chapter 4: Parameter dentiiation 4. ntrodution 4. Experimental Apparatu 4.. Tet Rig 4.. Senor v

7 Content 4... Torque 4... Speed Data Aquiition Analogue Signal Digital Signal Experimental Proedure nertia Frition Stine Damping Reult D.C. Motor Engine Flywheel Cluth Tranmiion Drivehat Wheel and Brake Dik Vehile 3 Chapter 5: Sytem Simulation ntrodution Working Point Simulation Complex v. Simpliied Model Eet o Gear Ratio Veriiation o Simulation Auray Publihed Reult Vehile Meaurement 37 Chapter 6: Controller Development 4 6. ntrodution 4 6. Control Strategy Etimator Deign Etimator Theory 4 vi

8 Content 6.3. Drivetrain Etimator Etimator Gain Etimator Perormane Control Sytem 48 Chapter 7: Controller Perormane ntrodution Control Gain Controller Perormane Analyi Comparion with Torque Rate Limiter Poible mprovement 57 Chapter 8: Conluion 58 Reerene 6 Appendix A: Drivetrain Model A. Appendix B: Tet Rig B. Appendix C: Drivetrain Parameter C. Appendix D: Direte Etimator D. vii

9 Lit o Figure Lit o Figure Figure..: Vehile Repone to a Rapid Change in Torque (Johanon (4)) Figure..: A Volkwagen Jetta Figure.3.: Flowhart o the Projet Outline Figure.3.: Complex v. Linear Model or a Torque Ramp rom Nm to Nm (Karlon, ) Figure.3.: Complex v. Linear Model or a Torque Ramp rom Nm to 8 Nm (Karlon, ) Figure.4.: PD Control (Lagerberg and Egardt, ) Figure.4.: PD Control with Torque Compenator (Lagerberg and Egardt, ) Figure.4.3: Simple Swithing Control (Lagerberg and Egardt, ) Figure.4.4: Modiied Swithing Control (Lagerberg and Egardt, ) Figure 3..: Drivetrain nertia Figure 3..: Cluth Stine Coeiient (Petteron, 996) Figure 3..: Drivehat with Baklah Figure 3..3: Longitudinal Fore on a Vehile Figure 3..4: Complex Model Blok Diagram Figure 3.3.: Simpliied Linear Drivetrain Model Figure 3.3.: Blok Diagram o the Simpliied Model Traner Funtion Figure 4..: Experimental Setup Figure 4..: Fully Aembled Tet Rig Figure 4..3: The Load Cell Setup Figure 4..4: Digital Magneti Pik up Speed Meaurement Figure 4.4.: nertia Tet Data Figure 4.4.: Frition Tet Data Figure 4.4.3: Stine Experiment Figure 4.4.4: Stine Tet Data (Maree, 5) Figure 4.5.: DC Motor Cloed Loop Bandwidth or Dierent Speed Amplitude (Ater Conradie, ) Figure 4.5.: Engine Torque Model Repone Figure 5.3.: Complex v. Simpliied Model ( Nm to 9 Nm Torque Ramp) Figure 5.3.: Complex v. Simpliied Model ( Nm to 7 Nm Torque Ramp) viii

10 Lit o Figure Figure 5.3.3: Vehile Aeleration Repone in Dierent Gear Figure 5.4.: Raw Aelerometer Tet Data Figure 5.4.: ECU Etimated Engine Torque Data Figure 5.4.3: Meaured v. Modelled Repone ( 4 Nm to 3 Nm in.5 eond) Figure 6..: Control Strategy Figure 6.3.: Engine Speed Meaurement Data Noie Figure 6.3.: Etimated v. Atual (Complex Model) Engine Speed (No Baklah) Figure 6.3.3: Etimated v. Atual (Complex Model) Wheel Speed (No Baklah) Figure 6.3.4: Etimated v. Atual (Complex Model) Engine Speed (Baklah) Figure 6.3.5: Etimated v. Atual (Complex Model) Wheel Speed (Baklah) Figure 6.3.6: Modelled Engine Speed with Noie v. Etimated Engine Speed Figure 6.4.: Shemati Overview o the Control Sytem Figure 7..: Controlled Aeleration Repone (No Baklah) Figure 7..: Repone Error v. Control Gain (No Baklah) Figure 7..3: Controlled Aeleration Repone (Baklah) Figure 7..4: Repone Error v. Control Gain (Baklah Figure 7.3.: Unontrolled v. Controlled Repone (No Baklah) Figure 7.3.: Unontrolled v. Controlled Repone (Baklah) Figure 7.3.3: Unontrolled v. Controlled Engine Speed Figure 7.3.4: Driver Demand v. Controlled Engine Torque Demand Figure 7.4.: Torque Rate Limiter v. Controller (No Baklah) Figure 7.4.: Torque Rate Limiter v. Controller (Baklah) ix

11 Lit o Table Lit o Table Table 4..: Experimental Proedure Table 4.5.: DC Motor Parameter Table 4.5.: Cluth Parameter Table 4.5.3: Tranmiion Parameter x

12 Gloary Gloary Nomenlature DAQ DC ECU PC PD RMS VSD Data Aquiition Diret Current Eletroni Control Unit ntegrated Powertrain Control Proportional ntegral Derivative Root Mean Squared Variable Speed Drive Symbol Г Ф α β ζ θ ρ ωn A b w Fa Fg Fr Ft g i Direte tate pae input matrix Direte tate pae tate matrix Hal the baklah angle Road gradient Damping ratio Angular diplaement (rad) Air denity Natural requeny Vehile maximum rontal area Viou rition oeiient Damping Vehile oeiient o drag Air drag ore Gradient ore Rolling reitane ore Trative wheel ore Gravitational aeleration Gear Ratio xi

13 Gloary nertia (kg.m ) k Stine K Control Gain L Etimator gain matrix Leng m Rv Rw rw T teng u v x y Engine tranport delay (lag) Vehile ma Proe noie ovariane Meaurement noie ovariane Wheel outer radiu Torque (Nm) Engine time ontant Sytem input Vehile veloity Sytem tate Sytem output Subript Cluth Cluth damping k Cluth tine Cluth rom to Cluth rom to 4 d Drivehat Flywheel L Load Drivehat t Tranmiion w Wheel dz Dead zone xii

14 Chapter ntrodution ntrodution. Projet Bakground With reent advane in automotive eletroni it ha beome poible to eletronially ontrol a vehile driveability. The term drivability reer to the driver pereption o the vehile repone to a ertain input. t enompae a range o apet inluding driver omort, vehile reponivene and handling. Driver omort ha beome a major area o reearh in the automotive indutry. Shunt and hule in a vehile drivetrain are two driveability related phenomena reponible or driver diomort. They are experiened a a harp jerk (hunt) ollowed by a erie o longitudinal oillation (hule) and are indued by a rapid hange in engine torque. The hunt and hule phenomena are learly viible in the vehile aeleration repone in Figure.. (Johanon, 4). n thi thei, driver omort i deined a a meaure o the amplitude o the hunt and hule. The ue o drive by wire throttle in modern day vehile enable the onboard eletroni ontrol unit (ECU) to manipulate the driver torque demand beore ending a revied torque demand ignal to the engine. n thi way ontroller an ue the engine a a torque atuator to the drivetrain. Drivetrain ontroller do exit in modern vehile, but many o them are imply lew rate limiter whih ilter any harp hange in driver torque demand into a lower, moother ignal. Although thee ytem do prevent hunt and hule they are oten over damped, and thi reult in a vehile that eem luggih and low. Thi redution in vehile reponivene i unaeptable, and the primary aim o thi projet wa to develop an eletroni ontrol ytem that would redue the hunt and hule without

15 Chapter ntrodution reduing the vehile reponivene to a torque demand. More peiially, the vehile reponivene i deined a the time taken or the vehile to reah teady tate aeleration in repone to a driver torque demand. Vehile Repone to a Rapid Change in Torque Aeleration (m/e ) Time (Seond) Figure..: Vehile Repone to a Rapid Change in Torque (Johanon (4)) n order to develop a ontroller, an aurate drivetrain model i required to predit the vehile repone to a torque input. Thi model an then be ued to deign and imulate the ontrol ytem perormane. n thi projet, the model reated wa baed on a.6 litre Volkwagen Jetta (Figure..). Figure..: A Volkwagen Jetta

16 Chapter ntrodution. Projet Objetive The primary objetive o thi thei are a ollow:. The development o an aurate working model or the imulation o the vehile longitudinal repone to variou torque input.. The deign and imulation o a drivetrain ontroller that will improve driver omort without reduing vehile reponivene..3 Projet Outline Figure.3. i a low hart o the variou tep ompleted in the projet. Mathematial Modelling: A mathematial model o the drivetrain wa derived. Parameter dentiiation: The drivetrain model parameter were identiied uing tet rig experiment, vehile meaurement data and reerening. Sytem Simulation: The identiied parameter were ued in the drivetrain model to imulate the vehile behaviour or variou torque input. The auray o the imulation wa veriied uing vehile meaurement data. Control Sytem Deign: A drivetrain ontroller wa deigned to redue hunt and hule, and it perormane wa imulated. Figure.3.: Flowhart o the Projet Outline 3

17 Chapter ntrodution.4 Chapter Overview The projet i doumented in the ollowing hapter: Chapter : Literature Study Chapter i a omprehenive literature tudy deribing the automotive drivetrain modelling and ontrol reearh that ha been ompleted over the lat ew year. Chapter 3: Drivetrain Modelling Two dierent drivetrain model are derived: A omplex non linear model or aurate imulation, and a impliied third order model or ue in ontrol ytem development. Chapter 4: Parameter dentiiation Thi hapter deribe the variou experimental proedure ollowed in order to obtain value or the drivetrain model parameter. The reult o thee proedure are then ummarized. Chapter 5: Vehile Simulation The model derived in Chapter 3 are ued to imulate the vehile repone to variou input and ondition. The perormane o the omplex non linear model i ompared to that o the linear third order model and the auray o the imulation i veriied through urther experimentation. Chapter 6: Controller Development A ontrol trategy and tate etimator are derived. The etimator perormane i then evaluated in more detail. Chapter 7: Controller Perormane The perormane o the drivetrain ontroller i analyzed in detail. Chapter 8: Conluion n thi hapter, the work ompleted i ummarized and everal reommendation are made or poible uture work on thi topi. 4

18 Chapter Literature Review Literature Review. ntrodution Thi hapter i a literature review o variou publiation on the ubjet o automotive drivetrain modelling and ontrol. The literature tudy wa divided into 3 etion and i ummarized in the ollowing paragraph:. Vehile Driveability.3 Drivetrain Modelling.4 Drivetrain Control Although muh work ha been done on the modelling o automotive drivetrain, the development o ontrol ytem or improved driveability i a relatively new ield.. Vehile Driveability Peron (4) i a literature urvey o approximately 8 paper related to variou driveability apet. The tudy inluded topi uh a utomer demand with regard to driveability, the meaurement o driveability, the variou type o automotive drivetrain, and dierent approahe taken toward improving vehile driveability. Peron deined driveability a the term whih deribe the driver omplex ubjetive pereption o the interation between driver and vehile. Variou ubjetive driveability indexe were examined and the need or an objetive, tandardized way o rating driveability wa identiied. Peron alo invetigated variou drivetrain model and onluded that impliied drivetrain model are ot eetive tool or developing drivetrain ontroller. Dierent drivetrain ontroller trategie were briely examined, and it wa onluded that it 5

19 Chapter Literature Review i in at poible to redue drivetrain oillation by uing an eletroni ontrol ytem. Johanon (4) developed a grading ytem or hunt and hule baed on an objetive mathematial analyi. Thi work ollowed the reearh o Peron (4) where the need or uh a ytem wa identiied. The grading ytem wa baed on eparate tet or hunt and hule and an overall grade (mark) wa alulated a the average between the hunt grade and the hule grade. The mathematial grading ytem orrelated well with the grade aigned by everal experiened proeional tet driver..3 Drivetrain Modeling Petteron (996) derived and validated a drivetrain model or a large truk. He preented three model o inreaing omplexity and then validated them through experimental teting. The irt model preented wa a linear model whih aumed a ti luth, a ti propeller hat and viou rition in the tranmiion and inal drive (dierential). The eond model added the luth lexibility to the original linear model. Finally, a more omplete non linear model wa derived whih inluded a luth model with a tati non linearity. The auray o eah o thee model wa veriied through vehile meaurement and their perormane were ompared. The ollowing onluion were made:. The main ontribution to driveline dynami rom driving torque to engine peed and wheel peed i the drivehat.. nluding luth dynami in the model doen t improve the model auray or drivetrain oillation requenie below 6 Hz. Karlon () preented two drivetrain model o a Volvo paenger ar; a linear third order model and omplex non linear model. The linear model exluded luth dynami and baklah. The omplex model aounted or both thee nonlinearitie. Karlon ompared the perormane o thee model or variou torque input. A imilar approah wa taken in the urrent tudy. Figure.3. how Karlon omparion o the omplex and linear model or a torque input ramp rom Nm to Nm in. eond. n thi enario the baklah region i ompletely avoided and the linear model perorm relatively well. 6

20 Chapter Literature Review Aleration (m/) Vehile Aeleration Repone Complex Model Linear Model Time (Seond) Figure.3.: Complex v. Linear Model or a Torque Ramp rom Nm to Nm (Ater Karlon, ) Figure.3. how the aeleration repone o Karlon omplex and linear model or a torque ramp rom Nm to 8 Nm in.5 eond. n thi enario the drivetrain pae through the baklah region and the linear model beome more inaurate. Aleration (m/) Vehile Aeleration Repone Complex Model Linear Model Time (Seond) Figure.3.: Complex v. Linear Model or a Torque Ramp rom Nm to 8 Nm (Ater Karlon, ) 7

21 Chapter Literature Review.4 Drivetrain Control The literature overing drivetrain ontrol i ar le extenive than or drivetrain modelling. Mot o the reearh done in thi ield ha been part o the ntegrated Powertrain Control (PC) projet between the Chalmer Univerity o Tehnology in Sweden, and the Volvo Corporation. Shunt and hule in an automotive drivetrain are predominantly due to the baklah in the drivetrain a well a the elatiity o variou drivetrain omponent. Karlon () deine baklah a the mall amount o mehanial lak between adjaent movable omponent. There are two major approahe that an be ued when deigning a drivetrain ontroller to prevent hunt and hule. The irt i to attempt to atively ontrol the drivetrain a it pae through the baklah region. Thi trategy require two ontrol mode; one or when the drivetrain i in the baklah region and another or when it i not. A non linear baklah model i inluded in thi type o ontroller. The eond approah i a linear approah whih doen t inlude a baklah model in the ontroller and imply treat the baklah a a diturbane. Thi type o ontroller i deigned to be robut enough to handle the diturbane. Lagerberg () i a literature tudy overing the modelling and ontrol o baklah in automotive drivehat. Three trategie are propoed to ontrol the drivetrain; a linear paive ontroller, a non linear paive ontroller and a nonlinear ative ontroller. With the linear paive approah, a linear ontroller wa deigned that wa robut enough to handle the baklah non linearity without inluding a model o it in the ontroller. The non linear paive trategy inluded a baklah model, but adopted a areul approah in the baklah region. The nonlinear ative approah inluded a baklah model and adopted the philoophy o getting out o baklah region a quikly a poible. Although thee trategie were identiied, they were not ued in imulation. Lagerberg and Egardt () propoed the ollowing ontroller; a tandard proportional, integral, dierential (PD) ontroller, a PD ontroller with a torque ompenator, a imple ative withing ontroller and a modiied withing ontroller. The tandard PD ontroller ued the load aeleration a the ontrolled variable. The PD ontroller with the torque oberver ued a irt order hat torque etimate ombined with the load aeleration a the ontrol ignal. The imple ative withing ontroller withed between two ontrol ignal; one when the drivetrain wa in the baklah region and another when it wa not. Thi ontroller exluded the engine dynami and hat lexibility. The modiied 8

22 Chapter Literature Review withing ontroller had a trategy imilar to that o the imple ative withing ontroller, but inluded the engine dynami in the baklah mode. The drivetrain repone to a torque tep or eah o the our ontroller i hown in Figure.4. through to.4.4. t i lear that the perormane o the dierent ontroller beome progreively better. Lagerberg and Egardt () onlude that although the withing ontroller learly out perorm the linear PD ontroller, i the baklah angle i overetimated thee withed ontroller beome untable. The withing ontroller are thereore not robut enough to implement in an atual vehile. Aeleration (m. ) Vehile Aeleration Repone Time (eond) Repone Demand Figure.4.: PD Control (Lagerberg and Egardt, ) Aeleration (m. ) Vehile Aeleration Repone Time (eond) Repone Demand Figure.4.: PD Control with Torque Compenator (Lagerberg and Egardt, ) 9

23 Chapter Literature Review Aeleration (m. ) Vehile Aeleration Repone Time (eond) Repone Demand Figure.4.3: Simple Swithing Control (Lagerberg and Egardt, ) Aeleration (m. ) Vehile Aeleration Repone Repone Demand Time (eond) Figure.4.4: Modiied Swithing Control (Lagerberg and Egardt, ) The previouly deribed withing ontroller require knowledge o when the drivetrain i in the baklah region o that the baklah ontrol mode an be implemented. Baklah tate etimator (oberver) were derived or thi purpoe in Lagerberg and Egardt (3). Non linear baklah model and Kalman iltering tehnique were ued to etimate the baklah ize and poition. Lagerberg and Egardt (4) i an experimental veriiation o the auray o thee etimate through vehile teting. t wa onluded that the etimate were o a high quality and that they were robut to modelling error.

24 Chapter 3 Drivetrain Modelling Drivetrain Modelling 3 3. ntrodution The powertrain i deined a everything that propel the ar. The drivetrain i the omplete powertrain exepting the engine. n other word, the drivetrain omprie o the lywheel, luth, tranmiion, drivehat and wheel. Figure 3.. how the major inertia, gear ratio and torque ued in the drivetrain model. n thi diagram, and θ are the lywheel inertia and angular poition. T i the engine torque tranmitted to the lywheel, T i the drivetrain torque ating on the tranmiion, and θ i the angular diplaement o the luth. t, θt, it and bt are the tranmiion inertia, angular poition, gear ratio and viou rition oeiient repetively. Td i the drivehat torque ating on the wheel, while w and θw are the ma moment o inertia and angular poition o eah wheel. Tt i the trative torque ating on the tyre. T θ θ θt, it θw T T Td Td Tt t w Flywheel Cluth Tranmiion Wheel Figure 3..: Drivetrain nertia

25 Chapter 3 Drivetrain Modelling n thi hapter a omplex non linear model o the drivetrain i derived. Thi model wa ued or advaned vehile imulation in Chapter 5. Next, the non linear model wa linearized and impliied. The primary purpoe o impliying the model wa that it would later impliy the proe o ontroller development. The ontroller would be developed uing thi impliied model and the eet o the additional non linear parameter would later be analyzed. The ontroller would then be tuned aordingly. 3. Complex Model 3.. Engine, Flywheel and Cluth Uing Newton eond law at the lywheel inertia, the ollowing model i obtained: θ = T T (3..) n order to develop a realiti drivetrain ontroller, engine dynami mut be onidered when alulating T. A deribed in Lagerberg and Egardt (), the engine i modelled a an ideal torque generator with irt order dynami and a imple tranport delay. n other word, the aumption i made that the ECU engine torque ontroller reult in the ollowing loed loop engine dynami: Le e T () = u() (3..) teng + The torque demand i u and the time ontant and tranport delay o the engine are teng and Le repetively. A maximum engine torque (T,max) i alo implemented. A deribed in Petteron (996), the luth onit o pring arranged irumerentially around a driven hub. The tine o thee pring determine how the torque i tranmitted rom the lywheel to the tranmiion. A hown in Figure 3.., the luth torional tine oeiient i k or a deletion angle rom to θ, and k or a deletion angle rom θ to the mehanial top (θ). The torque omponent due to the luth torional tine an thereore be deribed by the ollowing equation: θ = ( θ θt it) θ k i θ θ (3..3) T = θ k + ( θ θ) k i θ < θ < θ θ k + ( θ θ) k i θ θ

26 Chapter 3 Drivetrain Modelling Torque k k Mehanial top θ θ θ θ Mehanial top k Angular Diplaement k Figure 3..: Cluth Stine Coeiient (Petteron, 996) the luth damping oeiient i, the luth torque omponent due to damping i: T = θ (3..4) The overall luth torque T an thereore be written a: T = Tk + T (3..5) 3.. Tranmiion and Drivehat The gearbox and dierential orm one ompat tranmiion aembly in a rontwheel drive ar. A in Petteron (996), it wa aumed that the tranmiion wa ininitely ti when ompared to the luth. Due to the at that the drivetrain wa being modelled or longitudinal motion in a traight line, both ront wheel were propelled at the ame peed and the tranmiion ould thereore be modelled a a ingle inertia and gear ratio. Applying Newton eond law around the tranmiion inertia (Figure 3..): t θ t = T it bt θ t Td (3..6) The torque in the drivehat (Td) i a untion o the baklah in the drivetrain. Karlon () deine baklah a the mall amount o mehanial lak between adjaent movable omponent. Eah learane i relatively mall, but when they are added together the total baklah limit the ytem perormane. When the torque in the drivetrain hange diretion, the baklah hange ide and thi phenomenon make it important to inlude a model o the baklah to ahieve realiti imulation. 3

27 Chapter 3 Drivetrain Modelling There are three major baklah model in literature (Lagerberg, ): the dead zone model, the modiied dead zone model and the omplex model. The dead zone model i the mot ommonly ued, but i neverthele erroneou. damping i inluded in thi model it an be hown that the baklah an produe a pulling ore. The omplex model and modiied dead zone model have extra tate that prevent a pulling ore at ontat. The omplex verion relie on parameterized urve deription and i not very pratial. Karlon () howed that the modiied dead zone model oinide with the more omplex verion or realiti parameter etting, and it wa thereore ued to model the baklah in thi thei. Figure 3.. repreent a drivehat with baklah. The tine and damping oeiient o the drivehat are k and repetively, and α i hal the baklah angle. The modiied dead zone baklah model or the drivehat torque Td i repreented by the ollowing equation: θrel = θt θw k ( θrel α ) + θ rel i θrel > α Tdz = k ( θrel + α ) + θ rel i θrel < α (3..7) i θrel α i Tdz < andθrel > or i Tdz > andθrel< Td = Tdz otherwie α θt θw Td Td k Figure 3..: Drivehat with Baklah 3..3 Wheel and Vehile Ft i the trative ore on the moving wheel. Thi ore i onverted into a torque and Newton eond law i applied around the two driven wheel (Figure 3..): = T r F w θ w d w t (3..8) 4

28 Chapter 3 Drivetrain Modelling Figure 3..3 how the longitudinal ore ating on a vehile with ma m, veloity v and wheel radiu rw. Uing Newton eond law in the longitudinal diretion, the ollowing equation i obtained: m v = Ft Fa Fr Fg (3..9) Fa Ft Fr + Fg Figure 3..3: Longitudinal Fore on a Vehile Fa i the air drag ore. With the drag oeiient, maximum vehile ro etional area and air denity repreented by w, A, and ρ repetively, the drag ore i deribed by: Fa = w A ρ v (3..) Fr i the rolling reitane ore, and it wa modelled a in Wong (). The peed independent oeiient i r and the peed dependent oeiient i r. Fr = m g (r + r v ) (3..) Fg i the gradient ore on the vehile, and or a road with lope β it i deribed by: Fg = m g in( β ) (3..) The rition oeiient between a tyre and the road i a untion o the longitudinal lip o the tyre. Thi lip rate reult rom a redution o the eetive irumerene o the tyre due to the elatiity o the tyre rubber (Canuda De Wit et. al., 3). n thi drivetrain model, the tyre i aumed to roll without lip. The vehile and wheel veloitie are now related by: v = r θ Re writing (3..9) give: Ft = m v+ Fr + Fa + m g in( β ) Combining (3..8), (3..3) and (3..4) reult in: ( w + m r ) θ w = Td (Fr r + Fa r + m g r in( β )) (3..3) (3..4) (3..5) t i important to note that the tyre dynami reult in the majority o the drivetrain damping. Thee dynami are aounted or by inreaing the damping oeiient o the drivehat. 5

29 Chapter 3 Drivetrain Modelling 3..4 Summary n ummary, the three priniple dierential equation repreenting the drivetrain model are: θ = T T (3..) t θ t = T it Td bt θ t (3..6) ( w + m r ) θ w = Td (Fr r + Fa r + m g r in( β )) (3..5) A blok diagram overview o the model i hown in Figure More detailed (lower level) blok diagram an be ound in Appendix A. Flywheel Flywheel Aeleration Cluth Torque Fly wheel Speed Cluth Driver Torque Demand Torque Demand Engine Torque Engine Dynami Engine Torque Fly wheel Poition Fly wheel Poition Tranmiion Poition Cluth Torque T ranm ii on Tranmiion Aeleration Cluth Torque Tranmiion Speed Drivehat Torque Tranmiion Poition Drivehat with Baklah Tranmiion Speed Wheel Speed Tranmiion Poition Driv eha t Torque Wheel Wheel Aeleration Drivehat Torque Wheel Speed Wheel Aeleration Wheel Poition Load Fore Wheel Poition Load Fore Wheel Speed Load Fore Figure 3..4: Complex Model Blok Diagram 3.3 Simpliied Model For the purpoe o reduing the omplexity o the ontroller development proe, a impliied linear drivetrain model wa derived. Thi model would enable linear ontrol theory to be applied to the problem o drivetrain ontrol. The eet o the drivetrain non linearitie on the perormane o the ontroller wa later analyzed and the ontroller wa then tuned aordingly. The model ued in thi tudy wa preented in Fredrikon (999) and addition were made in Fredrikon et. al. (). Thi ame drivetrain model wa alo ued in Karlon (). Thi impliied model ignore engine dynami a well a rition in the drivetrain. The tranmiion inertia i aumed to be negligibly mall in omparion to that o 6

30 Chapter 3 Drivetrain Modelling the lywheel and wheel. Due to the tranmiion gear ratio, the drivehat ha a torque to tine ratio everal time that o the luth. The drivehat dynami are thereore aumed to dominate, and the luth i treated a being ti. Baklah i not aounted or. Flywheel Wheel Td/ it r Ft Td T it θ θw Tranmiion w Figure 3.3.: Simpliied Linear Drivetrain Model Uing the ame nomenlature a in Setion 3. and applying Newton eond law around the major inertia (Figure 3.3.), the ollowing equation reult: t d i T T = θ (3.3.) + = w t w t d i i k T θ θ θ θ (3.3.) t d w w F r T = θ (3.3.3) The longitudinal ore ating on the vehile are derived in the ame way a they were with the omplex model: )) in( r g m r F r (F T ) r m ( a r d w w β θ + + = + (3..5) Combining (3.3.), (3.3.) and (3..5), the dierential equation are written a: + = w t t w t t i i i i k T θ θ θ θ θ (3.3.4) t w t w t w T i i k + = θ θ θ θ θ (3.3.5) The term ha been ombined a a ingle inertia, and the trative torque Tt i deined a: ) r m ( w + ) in( r g m r F r F T a r t β + + = (3.3.6) The ollowing three tate variable are introdued; the torion in the drivehat, the wheel peed and the engine peed: 7

31 Chapter 3 Drivetrain Modelling 3 w w t X X ) i ( X = = = θ θ θ θ (3.3.7) The drivetrain model an now be written in tate pae orm a: + = t 3 t t t t t 3 T T X X X i i i k i k i X X X (3.3.8) Tt θ w θ e T GL() G() G() Figure 3.3.: Blok Diagram o the Simpliied Model Traner Funtion Converting the tate pae matrie o (3.3.8) into traner untion yield the ollowing (Figure 3.3.): i k k i k G () t t = (3.3.9) t t k i k i G () = (3.3.) t t L k i k i G () + + = (3.3.) 8

32 Chapter 3 Drivetrain Modelling The traner untion o engine torque to drivehat torque i: = t t t t i k k i i k i () G (3.3.) G() i a eond order ytem and it denominator thereore ha the orm: n n t t w i k k i + + = ω ζ (3.3.3) ζ and ωn are the damping ratio and natural requeny o the ytem repetively. The natural requeny o the drivetrain oillation (hule requeny) an thereore be written a: + = t n i k k ω (3.3.4) Thi how that the hule requeny i only dependent on the drivehat tine, lywheel inertia, gear ratio, wheel inertia, wheel radiu and vehile ma. 9

33 Chapter 4 Parameter dentiiation Parameter dentiiation 4 4. ntrodution Thi hapter deribe the proedure ued to identiy the drivetrain model parameter. The value hoen or eah omponent are then ummarized. Parameter value were obtained through experimentation on a tet rig, analyi o vehile meaurement data, or imply by reerening. Table 4.. indiate whih o the proedure wa ued or eah o the parameter. Table 4..: Experimental Proedure Baklah angle Experimentation Drivehat tine Experimentation Frition Experimentation Gear ratio Experimentation nertia Experimentation Drivetrain damping Vehile meaurement data Cluth tine Reerened Other Reerened 4. Experimental Apparatu 4.. Tet Rig A tet rig wa deigned and manuatured to mount the drivetrain on a large diret urrent (DC) eletri motor. A Siemen Simoreg (K type) variable peed

34 Chapter 4 Parameter dentiiation drive (VSD) wa ued to aurately ontrol the peed and torque output o the motor. Thi motor ontrol, oupled with aurate peed meaurement at everal plae along the drivetrain, generated data whih enabled the harateriation o everal drivetrain omponent. Figure 4.. i a hemati o the experimental etup and Figure 4.. how the aembled tet rig. (More detail on thi rig an be ound in Appendix B.) The motor wa mounted on our lea pring whih, i extended, would interet the motor axi. Thee mount thereore did not apply any moment on the motor rame, and any reation torque on the rame wa ened by the load ell whih wa onneted between the rame and tet bed. A hat with univeral joint wa ued to onnet the lywheel and motor. Thi hat wa the only omponent on the experimental etup that diered rom the atual drivetrain aembly in the ar. The lywheel tranerred torque rom thi hat to the tranmiion through the luth, whih wa permanently engaged. The tranmiion inluded the gearbox and dierential. The latter wa loked to enure that both wheel would be driven at the ame veloity. A ingle drivehat wa in turn onneted to the dierential, and the wheel and brake dik aembly wa onneted to thi drivehat. Drivehat Flywheel and Cluth Conneting Shat D.C. Motor Wheel Tranmiion Load ell Tahometer Tet bed with mount ndiate a peed meaurement point Figure 4..: Experimental Setup

35 Chapter 4 Parameter dentiiation Figure 4..: Fully Aembled Tet Rig 4.. Senor 4... Torque The load ell ued to meaure the torque wa attahed between the motor rame (whih wa ree to rotate) and the tet rig bae (Figure 4..3). The voltage output o the load ell wa diretly proportional to the ore exerted on it by the rotation o the motor rame. Sine the angle o rotation o the rame wa miniule, the ditane rom the entre o rotation to the load ell wa approximately ontant, and the voltage wa thereore alo diretly proportional to the reation torque on the motor. Figure 4..3: The Load Cell Setup

36 Chapter 4 Parameter dentiiation The load ell ignal wa paed through a bridge ampliier whih onverted it into a 5 to +5 V ignal. The load ell torque value and the torque ating on the drivetrain were related a ollow: T Load Cell = TDrivetrain + Motor θ Motor (4..) The load ell wa alibrated beore eah tet by hanging variou known weight on a.649 m torque arm. A linear aling ormula relating the meaured voltage and their relative torque value wa alulated. Thi ormula wa ued to ale all voltage meaured during teting to torque value Speed The motor peed wa meaured uing a tahometer whih provided a voltage proportional to it rotational peed. Thi voltage wa paed through the variable peed drive whih in turn onverted it into a to + V ignal. The ignal wa then onverted to peed (in rpm) uing the ollowing ormula: Speed (rpm) = V (4..) Other peed meaurement were made at the lywheel, ater the gearbox, and at the wheel uing tooth dik and digital magneti pikup (Figure 4..4). The dik were attahed along the axi o rotation o the drivetrain o that they rotated at the ame rate a the omponent at whih the meaurement wa being made. The pikup provided a 5 V digital (quare wave) ignal repreenting the riing and alling edge o the dik teeth. The proeing o thi ignal i urther deribed in Setion Figure 4..4: Digital Magneti Pik up Speed Meaurement 3

37 Chapter 4 Parameter dentiiation 4.3 Data Aquiition 4.3. Analogue Signal The load ell and tahometer provided analogue voltage ignal o the motor torque and peed repetively. Variou other analogue voltage waveorm were alo ued a input to the Siemen VSD, whih in turn enured that the motor ollowed thee ignal. A National ntrument PC 64 data aquiition (DAQ) ard wa ued to meaure all analogue input ignal a well a to generate all analogue output ignal. The program ued to ontrol the DAQ ard wa written in Borland C++ ine all the relevant National ntrument driver upported thi language. With regard to the hoen ampling time, Franklin et al. () ugget that a ample rate o to 4 time the ytem loed loop bandwidth hould be ued. Conradie () determined that the maximum loed loop bandwidth o the motor wa 5.33 Hz, and it wa thereore deided that a ample rate o Hz would be an appropriate hoie Digital Signal Thi orm o data aquiition wa applied to the peed meaurement made uing the digital magneti pik up. A National ntrument PC 66 ounter timer ard wa programmed (again uing Borland C++) to ount the number o pule produed by it onboard MHz oillator lok between ueive riing edge in the pik up ignal. The time taken or the toothed dik to rotate between ueive teeth ould then be alulated uing the orrelation: Time() = Count / (4.3.) For a dik with teeth the peed wa alulated uing: Time Speed(rpm) = (4.3.) 6 n thi way a peed reading wa obtained every 3. The maximum motor peed ued wa 5 rpm and the atet tooth period wa thereore. eond. Thi orreponded to a minimum o oillator lok pule per tooth period. The nature o the ounter operation i uh that 999, or pule ould be ounted depending on when ounting wa tarted. The wort reolution o the ounting operation wa thereore / =.5%. 4

38 Chapter 4 Parameter dentiiation 4.4 Experimental Proedure Thi etion deribe the experimental proedure ued to determine the variou drivetrain parameter. The reult o thee experiment are ummarized in Setion 4.5 or eah drivetrain omponent nertia The ma moment o inertia o eah drivetrain omponent around it axi o rotation wa determined by driving the motor with triangular peed untion (in other word ontant aeleration). To obtain the average aeleration, a regreion o the peed data wa perormed to obtain the gradient o the line. The reation torque on the load ell wa meaured and the average aeleration and torque value were then ubtituted into the ormula: T = (4.4.) θ Data rom an inertia experiment i hown in Figure The experiment wa ompleted on the tet rig with the drivetrain aembled up to and inluding the tranmiion. The DC motor wa aelerated and deelerated at approximately 4 rad.. nertia Tet Data 5 5 Torque Speed 4 Torque (Nm) 5 3 Speed (rpm) 5 5 Time (Seond) Figure 4.4.: nertia Tet Data 5

39 Chapter 4 Parameter dentiiation The inertia o the unloaded eletri motor wa alulated and the drivetrain wa then progreively aembled. A ull et o inertial experiment wa ompleted a eah new drivetrain omponent wa added. By ubtrating the ytem inertia obtained with the new omponent attahed rom the value obtained beore the omponent wa attahed, the inertia o the extra omponent ould be alulated. Frition in the ytem reulted in a reation torque meaurement that wa lightly too low during deeleration and lightly too high during aeleration. The mean torque rom eah peed wave period wa thereore alulated by averaging the abolute torque value meaured during aeleration and deeleration Frition t wa aumed that the rition in the drivetrain wa primarily due to the tranmiion. Thi rition wa aumed to be viou and i repreented by the ollowing equation: T = b θ (4.4.) The drivetrain wa driven at ontant peed and the average torque value were meaured. A traight line wa then itted to the torque veru peed data and the gradient o thi traight line wa ued a the rition oeiient b. Tet data rom a rition experiment on the tet rig i hown in Figure The drivetrain wa one again aembled up to and inluding the tranmiion (with 5 th gear engaged). A linear trendline ha been itted to the data. 5 4 Frition Tet Data Torque (Nm) Speed (rpm) Figure 4.4.: Frition Tet Data 6

40 Chapter 4 Parameter dentiiation Stine Figure i a hemati o how the tine experiment were onduted. One end o the drivetrain omponent wa ixed while a torque moment wa applied to the other end. The magnitude o the torional moment wa equal to the produt o the hanging weight and the torque arm length. Fixed End Torque Arm Hanging Ma Figure 4.4.3: Stine Experiment The angular deletion o the omponent wa meaured on the ree end and the experiment wa repeated or variou load. A linear regreion wa perormed on the torque veru deletion data and the gradient o the reulting traight line wa the tine o the omponent. Thi i deribed by: T k = (4.4.3) θ Tet data rom a luth tine experiment (Maree, 5) i hown in Figure Stine Tet Data Torque (Nm) Deletion (radian) Figure 4.4.4: Stine Tet Data (Maree, 5) 7

41 Chapter 4 Parameter dentiiation Damping Vehile behaviour wa imulated or hange in engine torque and gear hanging wa ignored. The luth wa thereore alway engaged and the aumption wa made that there wa no lip between the luth and lywheel. Thi allowed the drivetrain damping due to the luth, drivehat and tyre to be ombined a a ingle drivetrain damping oeiient () whih wa applied at the drivehat. The value ued i diued in Setion Reult 4.5. D.C. Motor Although the motor did not orm part o the drivetrain model, it wa neeary to haraterie it o a to identiy the drivetrain parameter. A previouly mentioned, the motor wa ontrolled by a VSD, and the unloaded loed loop DC motor ytem had the ollowing parameter: Table 4.5.: DC Motor Parameter Speed Range 55 rpm Maximum Aeleration 48 rpm. nertia.76 kg.m The bandwidth o a ytem i a meaure o the highet input ignal requeny that it an ollow aurately. Due to the at that the motor ha a large rotational inertia and limited power, it bandwidth will be highly dependent on the peed and amplitude o the oillation. Figure 4.5. how the loed loop bandwidth o the motor or variou peed and amplitude a obtained by Conradie (). For thee tet, the ut o requeny wa deined a the requeny at whih the magnitude o the output ignal wa 3 db le than the magnitude o the input ignal. 8

42 Chapter 4 Parameter dentiiation Bandwidth (Hz) Cloed Loop Bandwidth o the DC Motor rpm 4 rpm 5 rpm Speed (rpm) Figure 4.5.: DC Motor Cloed Loop Bandwidth or Dierent Speed Amplitude (Ater Conradie, ) 4.5. Engine The engine modelled wa a.6 litre normally apirated 4 ylinder park ignition engine. A previouly tated, it wa modelled a having irt order dynami and a time delay (lag). The maximum tranport delay wa alulated a the um o the ollowing delay:. The time taken or the preure rie to be tranported rom the throttle body to the inlet valve ater a tep hange in throttle poition;.5 eond. Thi wa alulated auming that the preure ront travelled at the peed o ound (345.9 m. at atmopheri ondition), and that the maniold length wa.5 m.. The time taken or a ingle inlet and a ompreion troke ( revolution). Thi i the time taken between the air entering the ylinder and the tart o the torque generation (ombution troke). The aumption i made that i the preure wave arrive at the inlet valve o a ylinder jut ater they have loed, it will immediately pa through the inlet valve o the next ylinder whih are open. The maximum total tranport delay or the engine i thereore.5 eond plu ull revolution. the engine peed i 3 rpm, the lag i: 9

43 Chapter 4 Parameter dentiiation L e = 3 = eond (4.5.) The time ontant o a irt order ytem i the time taken to reah 63.% o it teady tate value (Franklin et. al., ). t i aumed that i the torque generation begin at the tart o the ombution troke, the maximum torque point i reahed by the end o the troke. n other word ignoring the time delay, the teady tate torque value i reahed ater.5 revolution. The time ontant or the engine at 3 rpm i thereore: 3 Teng =.63.5 =.63 eond 6 The engine torque repone to a deired input i hown in Figure (4.5.) Engine Torque Model Repone Torque (Nm) Time (Seond) Firt Order Repone nput Demand Figure 4.5.: Engine Torque Model Repone The maximum engine torque or the engine (T,max) wa taken a 5 Nm Flywheel The lywheel inertia (), with the luth and preure plate attahed, wa meaured a.7 kg.m. To validate thi reult the lywheel wa approximated a a 3

44 Chapter 4 Parameter dentiiation irular dik. The analytial ormula or the ma moment o inertia o a irular dik i given in Hibbeler (997) a: = m r (4.5.3) The lywheel had an outer radiu o.5 m and a ma o 5. kg. Subtituting thee value into (4.5.) an inertia o =.7 kg.m i obtained. Thi value i almot idential to the experimentally determined value Cluth A deribed in Chapter, the luth ha a torional tine oeiient k or a deletion angle rom to θ, and k or a deletion angle rom θ to the mehanial top. Maree (5) howed that the luth had the ollowing parameter: Table 4.5.: Cluth Parameter θ (.94 rad) Maximum Deletion (Mehanial Stop) 4 (.443 rad) k N.m.rad k 67. N.m.rad Tranmiion The tranmiion onited o a 5 peed manual gearbox and dierential. Sine the drivetrain wa a ront wheel drive model, the gearbox and dierential ormed one ompat aembly. The value tabulated below are value meaured or the omplete aembly. Table 4.5.3: Tranmiion Parameter Gear Ratio (it) Frition Coeiient (bt) Gear.98.3 N.m..rad Gear N.m..rad Gear N.m..rad Gear N.m..rad Gear N.m..rad Revere..3 N.m..rad 3

45 Chapter 4 Parameter dentiiation The inertia o the tranmiion (t) wa meaured a.4 kg.m. Although thi inertia wa ar maller than the dominant lywheel and wheel inertia, it wa inluded in the omplex drivetrain model o a to provide a imple mean o inluding the luth propertie Drivehat The inertia o the drivehat wa meaured a.3 kg.m. Thi inertia wa negligibly mall when ompared to the inertia o the lywheel and wheel. The meaured drivehat tine (k) wa 64 N.m.rad, and vehile meaurement data (Momberg, 5) uggeted an overall drivetrain damping oeiient () o 9 N.m..rad. Thi value wa alulated by iteratively hanging the damping oeiient parameter in the drivetrain model until the modelled and meaured data orrelated well. Setion 5.4 urther diue the vehile teting proedure and ompare the meaured and modelled data in more detail. The baklah in the drivetrain wa meaured by loking one end o the drivetrain and meauring the angle o rotational movement (lak) on the other end. The total baklah angle (α) wa meaured a Wheel and Brake Dik The inertia o a ingle wheel and tyre wa meaured a.93 kg.m. Summating thi with the brake dik inertia (.7 kg.m ), the total wheel aembly inertia (w) wa. kg.m. The outer radiu o the wheel and tyre (rw) wa.3 m. Ater a erie o vehile tet on variou paenger ar, Wong () uggeted average rolling reitane oeiient o r =.36 and r = 5.8 x Vehile The vehileʹ maximum ro etional area (A) wa meaured a.5 m. The ma o the vehile (m) wa 4 kg. A typial value or the drag oeiient o modern day vehile (w) i.3 (White, 999). 3

46 Chapter 5 Sytem Simulation Sytem Simulation 5 5. ntrodution n thi hapter, a Simulink / Matlab model o the drivetrain (Appendix A) i generated to imulate the vehile repone to variou torque input. The reult are then analyzed and the auray o the imulation i veriied. Thi proe reult in an aurate vehile imulation tool whih an be ued in the development o a drivetrain ontroller. Appendix C provide a tabulated ummary o the variou parameter ued in the model. 5. Working Point Two enario were onidered in the vehile imulation and ontroller development. The irt wa the ituation where the engine torque output wa ramped rom Nm to 9 Nm in. eond. Thi ould our when the driver i initially aelerating lowly, and then hooe to rapidly inreae the aeleration o the vehile. t wa aumed that the throttle poition ollowed the pedal poition (or driver demand) intantaneouly. n thi enario the drivetrain doe not pa through the baklah region, and any hunt and hule that our i aued by the drivehat dynami. The eond enario onidered wa a torque ramp rom Nm to 7 Nm in. eond. Thi typially our when the throttle i loed (the ar i initially deelerating due to the reation torque by the road on the wheel), and the driver then hooe to rapidly aelerate the vehile. n thi ae, the drivetrain pae through the baklah region a the torque hange ign. 33

47 Chapter 5 Sytem Simulation The gear ratio in the tranmiion redue the engine peed and inreae the torque. For example, i the engine peed and torque are 3 rpm and Nm repetively, a gear ratio o.98 (irt gear) will reult in a wheel peed o 3/.98 = 3. rpm and a drivehat torque o.98 x = 98 Nm. Beaue higher gear ratio reult in higher drivehat torque value, the hunt and hule i wore in the lower gear. All imulation were thereore perormed in irt gear unle otherwie tated. 5.3 Simulation 5.3. Complex v. Simpliied Model The aeleration repone o the omplex and impliied model are ompared or both working point. The main dierene between the model are that the impliied model ignore the luth dynami and doe not aount or baklah, wherea the omplex model inlude both. Figure 5.3. ompare the two model when the drivetrain doen t pa through the baklah region (in other word or the Nm to 9 Nm torque ramp). Complex v. Simpliied Model 4 Aeleration (m/) 3 Complex Model Simpliied Model Torque Demand Torque Demand (Nm).5.5 Time (Seond) Figure 5.3.: Complex v. Simpliied Model ( Nm to 9 Nm Torque Ramp) 34

48 Chapter 5 Sytem Simulation t i lear that the repone i almot idential or both the omplex and imple model. Thi i to be expeted a the main dierene between the model i the baklah dynami, and the baklah region i ompletely avoided. Sine the linear model ignore the luth dynami and the omplex model inlude them, thee reult alo prove that the drivehat tine dominate. Thi i explained by the at that the torque to tine ratio with the irt gear gearing i higher in the drivehat than in the luth. t i thereore lear that i the region around negative torque i avoided, the impliied model repreent the vehile behaviour well. When the eond enario i imulated (a torque input rom Nm to 7 Nm), the linear model doen t ompare a well to the omplex model. Thi i due to the at that the drivetrain now pae through the baklah region (Figure 5.3.). t i intereting to note that the requeny o the oillation (hule) i till the ame or both model (.6 Hz in irt gear). Thi one again onirm that the drivehat tine i the dominant drivetrain tine. Complex v. Simpliied Model Aeleration (m/) Complex Model Simpliied Model Torque Demand 6 4 Torque Demand (Nm) Time (Seond) Figure 5.3.: Complex v. Simpliied Model ( Nm to 7 Nm Torque Ramp) 35